Algorithms for Forward and Backward Solution of the Fokker-Planck Equation in the Heliospheric Transport of Cosmic Rays

Wawrzyńczak, A.; Modzelewska, Renata; Gil, Agnieszka

doi:10.1007/978-3-319-78024-5_2

Cited by 3 publications

(8 citation statements)

References 12 publications

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“…In that case the PTE should be rewritten in the form of the FPE (for details see, e.g. [14], [15]), as:…”

Section: The Algorithm For Numerical Solution Of Thementioning

confidence: 99%

“…(1) to the solution of the set of stochastic ordinary differential equations (SDEs) being the exact equivalence of the FPE (e.g., [16]). Details of this procedure are given in ( [14], [15] and references therein). Accordingly, the transport of CR in the 2D heliocentric coordinate system, in which r = (r, θ), can be described by the following SDEs:…”

Section: The Algorithm For Numerical Solution Of Thementioning

confidence: 99%

“…The stochastic terms contain an element d W which is the increment of Wiener process guiding the stochastic motion of pseudoparticles in given dimension. B ij , (i, j = r, θ) is a matrix given in [15]. We discretize the set of Eqs.…”

Section: The Algorithm For Numerical Solution Of Thementioning

confidence: 99%

“…The simulation of pseudoparticles motion is done by the function SEQ() given in lines 12-36. The pseudoparticle initial position and rigidity is set in line 16-18; its position is changed accordingly to the equation 2 (lines [15][16][17][18][19] including the Wiener process defined in line 20-21. The initial weight is set in line 19, while it change is set in line 31.…”

Section: Julia Parallel Code For the Solution Of Fpementioning

confidence: 99%

“…To obtain the reliable results and achieve the satisfactory statistics in each 2D heliosphere bin we have to run at least 2 millions of pseudoparticles from the heliosphere boundary. In this paper we do not focus on the results of the physical model, interested reader we refer to the following papers [14], [15]. Here we would like to test the performance of above presented parallel Julia code on the HPC cluster.…”

Section: A Julia Performancementioning

confidence: 99%

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Parallelizing the code of the Fokker-Planck equation solution by stochastic approach in Julia programming language

Wawrzyńczak

2018

Annals of Computer Science and Information Systems

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Presenting a reliable physical simulation requires very often use of the supercomputers and models run for many days or weeks. The numerical computing is divided into two groups. One uses highly efficient low-level languages like Fortran, C, and C++. The second applies high-level languages like Python or Matlab, being usually quite slow in HPC applications. This paper presents the application of the relatively new programming language Julia, advertised as the as "a high-level, highperformance dynamic programming language for numerical computing". We employ Julia is to solve the Fokker-Planck equation by the stochastic approach with the use of the corresponding set of ordinary differential equations. We propose the method of parallelizing the algorithm with use of the distributed arrays. We test the speedup and efficiency of the given code with use of the cluster set at theŚwierk Computing Centre and show that Julia is capable of achieving a good performance.

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“…In that case the PTE should be rewritten in the form of the FPE (for details see, e.g. [14], [15]), as:…”

Section: The Algorithm For Numerical Solution Of Thementioning

confidence: 99%

Section: The Algorithm For Numerical Solution Of Thementioning

confidence: 99%

Section: The Algorithm For Numerical Solution Of Thementioning

confidence: 99%

Section: Julia Parallel Code For the Solution Of Fpementioning

confidence: 99%

Section: A Julia Performancementioning

confidence: 99%

See 3 more Smart Citations

Parallelizing the code of the Fokker-Planck equation solution by stochastic approach in Julia programming language

Wawrzyńczak

2018

Annals of Computer Science and Information Systems

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Approximate Bayesian Computation for Estimating Parameters of Data-Consistent Forbush Decrease Model

Wawrzyńczak

Kopka

2018

Entropy

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Realistic modeling of complex physical phenomena is always quite a challenging task. The main problem usually concerns the uncertainties surrounding model input parameters, especially when not all information about a modeled phenomenon is known. In such cases, Approximate Bayesian Computation (ABC) methodology may be helpful. The ABC is based on a comparison of the model output data with the experimental data, to estimate the best set of input parameters of the particular model. In this paper, we present a framework applying the ABC methodology to estimate the parameters of the model of Forbush decrease (Fd) of the galactic cosmic ray intensity. The Fd is modeled by the numerical solution of the Fokker–Planck equation in five-dimensional space (three spatial variables, the time and particles energy). The most problematic in Fd modeling is the lack of detailed knowledge about the spatial and temporal profiles of the parameters responsible for the creation of the Fd. Among these parameters, the diffusion coefficient plays a central role. We employ the ABC Sequential Monte Carlo algorithm, scanning the space of the diffusion coefficient parameters within the region of the heliosphere where the Fd is created. Assessment of the correctness of the proposed parameters is done by comparing the model output data with the experimental data of the galactic cosmic ray intensity. The particular attention is put on the rigidity dependence of the rigidity spectrum exponent. The proposed framework is adopted to create the model of the Fd observed by the neutron monitors and ground muon telescope in November 2004.

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Statistical Error for Cosmic Rays Modulation Evaluated by SDE Backward in Time Method for 1D Model

Mykhailenko

Bobík

2022

Fluids

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The propagation of cosmic rays through the heliosphere has been solved for more than half a century by stochastic methods based on Ito’s lemma. This work presents the estimation of statistical error of solution of Fokker–Planck equation by the 1D backward in time stochastic differential equations method. The error dependence on simulation statistics and energy is presented for different combinations of input parameters. The 1% precision criterion in mean value units of intensity standard deviation is defined as a function of solar wind velocity and diffusion coefficient value.

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Algorithms for Forward and Backward Solution of the Fokker-Planck Equation in the Heliospheric Transport of Cosmic Rays

Cited by 3 publications

References 12 publications

Parallelizing the code of the Fokker-Planck equation solution by stochastic approach in Julia programming language

Parallelizing the code of the Fokker-Planck equation solution by stochastic approach in Julia programming language

Approximate Bayesian Computation for Estimating Parameters of Data-Consistent Forbush Decrease Model

Statistical Error for Cosmic Rays Modulation Evaluated by SDE Backward in Time Method for 1D Model

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